Semisimple Lie algebra

Here, the connection or the gauge potential A assuming values in a(n irreducible) representation R of the compact, semisimple Lie algebra g of the Lie group G is of the form... 

In particular, all so-called semisimple Lie algebras satisfy this condition. The bilinear form tr(adxudy)(or Re tr adxady) is called the Killing form. Let us denote it simply by (X, Y). Semisimple Lie algebras are characterized by the fact that the form (X, y) is nondegenerate. 

All basic effects connected with the root system of semisimple Lie algebras manifest themselves fully on the example of a semisimple Lie algebra sl(n, C). For this reason and for the reader s convenience, we will illustrate the basic properties of the root system on the example of this algebra. After the acquaintance with this example, the interested reader will be able to confidently and without difficulty on root systems in any other semisimple Lie algebra. 

Up to this moment we have dealt with complex semisimple Lie algebras. But an important role is also played by various real subalgebras contained in complex algebras. One of them is especially remarkable, since the corresponding Lie group is compact. 

Let Go be a certain real form of a complex semisimple Lie algebra G. Then any element of the algebra G can be uniquely represented in the form X - -iYj where X,Y 6 Go. This decomposition of the algebra G gives rise to a natural involution 6 which maps the algebra G into itself. Namely a X- -iY) = X—iY. This involution depends on the subalgebra Go and possesses the following obvious properties ... 

Lemma 1.4.2. If Go is the real form of the semisimple Lie algebra G, then two above-mentioned forms coincide (up to the nonzero factor). 

Some maximal linear commutative algebras of functions on semisimple Lie algebras were later constructed in [16] by Bogoyavlensky. 

The next class of Lie algebras, in some sense rather close to the class of semisim-ple algebras, is an extension of semisimple Lie algebras by means of linear representations of minimal dimension. Recall some definitions. 

Remark In the proof of the equality ind 0 + ind G for semisimple Lie algebras, we may use the Rais es formula [205] for the index of a semidirect product. But in this case, as before, the Lie algebra is to be found, which makes up the crucial point in the proof of the formula for the index of the Lie algebra G. The theorem follows. 

This assertion is a generalization of the results deduced by Dao Chong Thi and Brailov for complete involutive sets on singular orbits of semisimple Lie algebras. 

Compact real form Gc of a semisimple Lie Algebra. Fomenko and Mishchenko, see Theorem 4.1.1 and [89], [91], [93j. Compact series. Polynomials. Dao Chong Thi (proof with lacunas). Brailov, Theorem 4.1.2. Compact series. Polynomials. No such orbits. 

Semisimple Lie algebras. The new sets of functions (see on the right) differ from the familiar ones, see items 1-3 above. 

K 0p V, where K is an arbir trary semisimple Lie algebra (real or complex), p an arbitrary linear representation (reducible or irreducible), dim V being greater than dim K, and in the decomposition of the representation p into irreducible components trivial summands are absent. Brailov, Pevtsova [106], [111]. All the functions of these complete involutive sets are linear. 

G 0ad Of where G is a semisimple Lie algebra (complex or real). This algebra is a particular case of Lie algebras of the type G 0 A (see item 14 above). To this end, one should take A = R[x]/(x ) (or in the complex case A = C[ ]/(rr )). Bolsinov [277]. His complete involutive sets on these algebras differ from those constructed earlier by lYofi-mov and Brailov (see item 14 above). 

Not to burden the presentation, we dwell only on one series of such examples associated with various equations of motion of a rigid body (in the multidimensional situation). To demonstrate our general method, we elucidate more or less comprehensively the procedure of studying the maximal linear commutative algebra of polynomials performed in Theorem 4.1.1 for a complex semisimple Lie algebra G. 

Let V be the maximal linear commutative algebra of polynomials, on orbits of a semisimple Lie algebra G, presented in Theorem 4.1.1 and in Proposition 4.1.1 (Sec. 1.2). We will now describe the subspace of quadratic Hamiltonians contained in 7. 

Thus, we have completely described the subspace of quadratic functionals in the maximal commutative linear algebra V of polynomial on the orbit of general position in a semisimple Lie algebra. 

Integrability of these equations on singular semisimple orbits in semisimple Lie algebras was later proved by Dao Chong Thi [39] and Brailov [206], [208]. 

Theorem 4.2.6. Let p G G be a linear operator, on a semisimple Lie algebra G, self-conjugate with respect to the Killing form. The Euler equation X = [X, pX] is Hamiltonian simultaneously with respect to both Poisson brackets (the element a is a covector of general position), and, a if and only if p [Pg.217]

Compatible Poisson brackets on Lie algebras were analyzed in the paper by Reyman [117], where such brackets appeared from infinite-dimensional graduated Lie algebras and were applied to the study of the various generalizations of Toda chains. In the same paper [117], Reyman pointed out the Hamiltonian property of the Euler equations for the shifts of invariants of semisimple Lie algebras indicated earlier in the paper [247]. 

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